#6586119

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1530 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,8)(4,6)(5,7)(9,10,12,14)(11,15,16,13)(17,18)(19,26)(20,23)(21,24)(22,25)(27,30)(28,29)(31,32);(1,3,5,4)(2,6,7,8)(17,18)(19,27)(20,29)(21,24)(22,32)(23,28)(25,31)(26,30);(9,11,12,16)(10,13,14,15)(17,19,21,20)(18,23,24,26)(22,30,31,28)(25,29,32,27);(1,4,5,3)(2,6,7,8)(19,28)(20,30)(22,31)(23,27)(25,32)(26,29);(17,20,21,19)(18,23,24,26)(22,28,31,30)(25,29,32,27)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=g,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6493315);